GIFTS clear sky fast model, its adjoint, & the neglected reflected term MURI Hyperspectral Workshop Madison WI, 2005 June 7 bob knuteson, leslie moy, dave tobin, paul van delst, hal woolf

Outline of Talk Fast Model: Development & Status Tangent Linear, Adjoint: Development & Status Surface Reflected Term: Work in Progress

Compute monochromatic Fast Model Production Flowchart: Profile Database Fixed Gas Amounts Spectral line parameters Lineshapes & Continua Layering, l Compute monochromatic layer-to-space transmittances Reduce to sensor’s spectral resolution Effective Layer Optical Depths, keff Convolved Layer-to-Space Transmittances, tz (l) Fast Model Predictors, Qi Fast Model Regressions Fast Model Coefficients, ci

RMS Error: water lines Red=before, Blue=after Why the improvement? Mainly from SVD regression & Optical Depth Weighting.

RMS Error: water continuum Red=before, Blue=after Why the improvement? Mainly from regressing nadir only Optical Depths and applying constant factors to off-nadir values.

MURI model w/ new regressions OPTRAN, AIRS 281 channel set ------- GIFTS NeDT@296K ------- OSS RMS upper limit* Dependent Set Statistics: RMS(LBL-FM) Yr 2002 model MURI version MURI model w/ new regressions AIRS model c/o L. Strow, UMBC OSS model c/o Xu Liu, AER, Inc. OPTRAN, AIRS 281 channel set c/o PVD

Profile of temperature, User Input: User Output: Profile of temperature, dry gases, water vapor at 101 levels Profile perturbation of temperature, ozone, water vapor at 101 levels Use to adjust initial profile Forward Model: Adjoint Model: Layer.m - convert 101 level values to 100 layer values Predictor.m - convert layer values to predictor values Calc_Trans.m - using predictors and coefficients calculate level to space transmittance Trans_to_Rad.m - calculate radiance Layer_AD.m - layer to level sensitivities Predictor_AD.m - level to predictor sensitivities Calc_Trans_AD.m - predictor to transmittance sensitivities Trans_to_Rad_AD.m - transmittance to radiance sensitivities Ill posed User Output: User Input: Compare to observations Radiance Spectrum Radiance Spectrum perturbation

Simple Example: One Line Forward Model Forward (FWD) model. The FWD operator maps the input state vector, X, to the model prediction, Y, e.g. for predictor #11: Tangent-linear (TL) model. Linearization of the forward model about Xb, the TL operator maps changes in the input state vector, X, to changes in the model prediction, Y, iteration Or, in matrix form:

TL testing for Dry Predictor #6 (T2) vs Temp at layer 44 TL testing for Dry Predictor #6 (T2) vs Temp at layer 44. * TL results must be linear. * TL must equal (FWD-To) at dT=0. TL results = blue, FWD-T0 results = red Difference between TL and FWD Input Temperature at Layer 44 were varied 25%.

TL testing for Dry Predictor #6 vs Temp at all layers TL testing for Dry Predictor #6 vs Temp at all layers. Similar plots made for each subroutine’s variables. D(dry.pred#6) Layer no. D(temp), %

Adjoint (AD) model. The AD operator maps in the reverse direction where for a given perturbation in the model prediction, Y, the change in the state vector, X, can be determined. The AD operator is the transpose of the TL operator. Using the example for predictor #11 in matrix form, Expanding this into separate equations:

Adjoint code testing for Dry Predictor #6 vs Temperature layer Adjoint code testing for Dry Predictor #6 vs Temperature layer. AD - TLt residual must be zero. Similar plots are produced for every subroutine’s variables. x 10 -18 AD - TLt residual Output variable layer Input variable layer

Clear Sky Top of Atmosphere Radiance I TOA = I atmos + I surf emiss + I surf reflect = I atmos + Ttoa B(tempsurf ) surf + Ttoa Fluxsurf Reflectivity I atmos I surf emiss I surf reflect r, r current fast model we write the expression more explicitly below I surf reflect = Ttoa   I(i,i) cos(i) sin(i) d(i) d(i)  BDRF(r,r: i,i) This Term is often ignored because Refl < 10%. IF the term is calculated accurately enough, it can be exploited to derive surf and hence Tsurf

Approximations made & Their Associated Errors I surf reflect = Ttoa   I(i,i) cos(i) sin(i) d(i) d(i)  BDRF(r,r: i,i) Approx.1: Lambertian surface (reflection is independent of incident angle): BDRF = R(r,r) = 1- surf (r,r) Approx.2: Low Order Gaussian Quadrature technique for calculating flux # quadrature points needed? which table to use? (Abramowitz and Stegun, 1972) Approx.3: Resolution Reduction SRF  {Ttoa  2   I( )  d }  {SRF  Ttoa }  {SRF  2   I( )  d } Slow, smoothing varying radiance as a function of cos theta Approx.4: Calculating Downwelling Radiance from Upwelling Fast Model SRF  I() (using LBLRTM)  (T GIFTS layer to space convolved) B(templayer)

Downwelling flux = 0 0 I(,) cos() sin() d d Expanding on Approx. 2: 2 /2 Downwelling flux = 0 0 I(,) cos() sin() d d = 2 0 I() cos() sin() d substituting  = cos (), = 2 0  I() d Diffusivity approximation, Low Order Gaussian Quadrature technique 0  I() d =  wi I(i ) p.921, Abramozwitz & Stegun, 1972 n=1,  0.5 I(=48) n=2,  0.2 I(1=69°) + 0.3 I(2 =32°) In contrast to the 2-stream model application: -1 I() d =  wi I(i ) p.916, Abramozwitz & Stegun, 1972 n=1,  I(=54.73) /2 1 1 n i=1 Azmuthially symmetric to get rid of the phi 1 n i=1

Approx 2: Error in Gaussian Quadrature Approximations (difference from using 4 points) Using two points Difference, W/(cm2 cm-1 ster) Differences in window regions Using one point

Approx. 3: Convolution Error Product of Convolution Minus Convolution of Products Difference, W/(cm2 cm-1 ster)

Errors from 1 Point Gaussian Quad & Convolution Approximations 1 point Gauss. Qaud. Both approx. Why convolution error is a positive bias?

Errors from 2 Point Gaussian Quad & Convolution Approximations Both approx. 2 point Gauss. Qaud.

Approx. 4: Using Fast Model Upwelling Level-2-space Transmissivity to calculate Downwelling Radiance TOA radiance rad = rad + 0.5 (ba+bb) (1-Tb/ Ta ) Ta layer trans level 2 space layer radiance emission Ta Tb BOA radiance rad = rad + 0.5 (ba+bb) (1-Tb/ Ta ) (T1/ Tb) T1 level 2 ground Close up of the window region Differences lblrtm - From Tran(lev2space) Comparison of Downwelling Radiance Built in directionality from Tran(lev2space) from lblrtm

Accomplishments: • Coefficients promulgated 2003. Reproduce and Upgrade existing GIFTS/IOMI Fast Model • Coefficients promulgated 2003. • Greatly improved the dependent set statistics (esp. water vapor). • Water continuum regression made at nadir and applied to all angles. • SVD regression and optical depth weighting incorporated. • Written in flexible code with visualization capabilities. Under CVS control. Write the Corresponding Tangent Linear and Adjoint Code • Tested to machine precision accuracy. • User friendly “wrap-around” code complete. • Transferred code to FSU. Investigate Surface Reflected Radiance • Great improvement with two point Gaussian Quadrature (over 1 point). • Convolution order causes large errors – may be overcome with regression algorithm? • Depending on the application (micro-window or on/off line) using upwelling transmissivity for downwelling radiance may be reasonable.